Ralph Blumenhagen et al- GUTs in Type IIB Orientifold Compactifications

8/3/2019 Ralph Blumenhagen et al- GUTs in Type IIB Orientifold Compactifications

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ESI The Erwin Schrodinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, Austria

GUTs in Type IIB Orientifold Compactifications

Ralph Blumenhagen

Volker Braun

Thomas W. Grimm

Timo Weigand

Vienna, Preprint ESI 2108 (2009) February 11, 2009

Supported by the Austrian Federal Ministry of Education, Science and CultureAvailable via http://www.esi.ac.at


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MPP-2008-144DIAS-STP 08-15

Bonn-TH 2008-15SLAC-PUB-13466

GUTs in Type IIB Orientifold Compactifications

Ralph Blumenhagen1, Volker Braun2, Thomas W. Grimm3, and Timo Weigand4

1 Max-Planck-Institut fur Physik, Fohringer Ring 6,80805 Munchen, Germany

2 Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland3 Bethe Center for Theoretical Physics and

Physikalisches Institut der Universitat Bonn, Nussallee 12,53115 Bonn, Germany

4 SLAC National Accelerator Laboratory, Stanford University,

2575 Sand Hill Road, Menlo Park, CA 94025, USA

[email protected], [email protected],

[email protected], [email protected]

PACS numbers: 11.25.-w, 11.25.Wx

Abstract

We systematically analyse globally consistent SU(5) GUT models on inter-secting D7-branes in genuine Calabi-Yau orientifolds with O3- and O7-planes.

Beyond the well-known tadpole and K-theory cancellation conditions thereexist a number of additional subtle but quite restrictive constraints. For therealisation of SU(5) GUTs with gauge symmetry breaking via U(1)Y flux wepresent two classes of suitable Calabi-Yau manifolds defined via del Pezzotransitions of the elliptically fibred hypersurface P1,1,1,6,9[18] and of the Quin-tic P1,1,1,1,1[5], respectively. To define an orientifold projection we classify allinvolutions on del Pezzo surfaces. We work out the model building prospectsof these geometries and present five globally consistent string GUT models indetail, including a 3-generation SU(5) model with no exotics whatsoever. Wealso realise other phenomenological features such as the 10105H Yukawa cou-pling and comment on the possibility of moduli stabilisation, where we find

an entire new set of so-called swiss-cheese type Calabi-Yau manifolds. It isexpected that both the general constrained structure and the concrete modelslift to F-theory vacua on compact Calabi-Yau fourfolds.


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Contents

1 Introduction 4

2 Orientifolds with Intersecting D7-Branes 10

2.1 Intersecting D7-Branes With Gauge Bundles . . . . . . . . . . . . . . 11

2.2 Tadpole Cancellation for Intersecting D7-Branes . . . . . . . . . . . . 16

2.3 The Massless Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Matter Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2 Matter Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 F- and D-Term Supersymmetry Constraints . . . . . . . . . . . . . . 23

3 SU(5) GUTs and Their Breaking 253.1 Georgi-Glashow SU(5) GUT . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Breaking SU(5) to SU(3) SU(2) U(1) . . . . . . . . . . . . . . . 313.3 Summary of GUT Model Building Constraints . . . . . . . . . . . . . 37

4 Del Pezzo Transitions on P1,1,1,6,9[18] 38

4.1 Del Pezzo Surfaces and Their Involutions . . . . . . . . . . . . . . . . 39

4.2 The Geometry of Del Pezzo Transitions ofP1,1,1,6,9[18] . . . . . . . . . 43

4.3 Orientifold of An Elliptic Fibration Over dP2 . . . . . . . . . . . . . 47

4.4 Orientifold of An Elliptic Fibration Over dP3 . . . . . . . . . . . . . 534.5 The Swiss-Cheese Property . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 D-Term Conditions For D7-Branes on Del Pezzo Surfaces . . . . . . . 58

5 A GUT Model on M(dP9)

2

260

5.1 The Chiral Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 D3-Brane Tadpole and K-Theory Constraints . . . . . . . . . . . . . 62

5.3 D-Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4 Globally Consistent Model . . . . . . . . . . . . . . . . . . . . . . . . 65

6 GUT Model Search 73

6.1 A 3-Generation GUT Model on M(dP8)2

2 . . . . . . . . . . . . . . . . . 73

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6.2 A GUT Model on M(dP9)

3

3 . . . . . . . . . . . . . . . . . . . . . . . . 80

7 GUTs on Del Pezzo Transitions of the Quintic 85

7.1 Del Pezzo Transitions of the Quintic . . . . . . . . . . . . . . . . . . 877.2 A GUT Model Without Vector-Like Matter . . . . . . . . . . . . . . 92

7.3 A Three-Generation Model With Localised Matter on Q(dP9)4

. . . . . 96

8 Comments on Moduli Stabilisation 98

9 Conclusions 101

A Involutions on Del Pezzo Surfaces 104

A.1 Del Pezzo Surfaces of High Degree . . . . . . . . . . . . . . . . . . . . 104

A.2 Involutions on the Projective Plane . . . . . . . . . . . . . . . . . . . 104

A.3 Involutions on the Product of Lines . . . . . . . . . . . . . . . . . . . 106

A.4 Blow-up of the Projective Plane . . . . . . . . . . . . . . . . . . . . . 107

A.5 Blow-up of the Projective Plane at Two Points . . . . . . . . . . . . . 109

A.6 Blow-up of the Projective Plane at Three Points . . . . . . . . . . . . 112

A.7 The Weyl Group and The Graph of Lines . . . . . . . . . . . . . . . . 115

A.8 Minimal Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A.9 Blow-ups of Minimal Models . . . . . . . . . . . . . . . . . . . . . . . 122

A.10 Explicit Realisations . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

B Cohomology of Line Bundles over del Pezzo Surfaces 126

C Cohomology of Line Bundles On Rational Elliptic Surfaces 129

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1 Introduction

The LHC experiment is widely expected not only to confirm the existence of the Higgs

particle as the last missing ingredient of the Standard Model of Particle Physics, butalso to reveal new structures going far beyond. As experiments are proceeding intothis hitherto unexplored energy regime, string theory, with its claim to represent theunified theory of all interactions, will have to render an account of its predictionsfor physics beyond the Standard Model. Clearly, these depend largely on the valueof the string scale Ms, the most dramatic outcome corresponding to Ms close to the

TeV scale. While this is indeed a fascinating possibility, in concrete string models itoften leads to severe cosmological issues such as the cosmological moduli problem.In this light it might be fair to say that a more natural (but also more conservative)scenario involves a value of Ms at the GUT, Planck or intermediate scale.

During the last years, various classes of four-dimensional string compactificationswith N= 1 spacetime supersymmetry have been studied in quite some detail (seethe reviews [1, 2, 3, 4,5] for references). From the viewpoint of realising the Mini-mal Supersymmetric Standard Model (MSSM) and some extension thereof the bestunderstood such constructions are certainly the perturbative heterotic string andType IIA orientifolds with intersecting D6-branes. On the contrary, as far as modulistabilisation is concerned Type IIB orientifolds with O7- and O3-planes look verypromising. The combination of three-form fluxes and D3-brane instantons can sta-bilise all closed string moduli [6] even within the solid framework of (conformal)Calabi-Yau manifolds where reliable computations can be performed. Moreover, su-persymmetry breaking via Kahler moduli mediation and the resulting structure ofsoft terms bear some attractive features and have been studied both for the LARGEvolume scenario [7, 8] with an intermediate string scale and for a GUT scenario with

the string scale at the GUT scale [9,10].

These considerations are reason enough to seriously pursue model building withintype IIB orientifolds. The observation that the MSSM gauge couplings appear tomeet at the GUT scale furthermore suggests the existence of some GUT theory athigh energies. GUT gauge groups such as SU(5) and SO(10) appear naturally in

string theories based on gauge group E8 like the heterotic string. On the other hand,it has become clear that for perturbative orientifolds with D-branes, exceptionalgauge groups and features like the spinor representations of SO(10) do not emerge.For SU(5) D-brane models, by contrast, the gauge symmetry and the desired chiralmatter spectrum can be realised, a fact welcome in view of the described progress in

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Type IIB moduli stabilisation. Still, at first sight there appears a serious problem inthe Yukawa coupling sector. The 10105H Yukawa coupling violates global pertur-bative U(1) symmetries which are the remnants of former U(1) symmetries rendered

massive by the Stuckelberg mechanism [11]. As a consequence of these considerationsit is sometimes argued that the natural context for Type II GUT model building isthe strong coupling limit, where the crucial couplings in question are not perturba-tively forbidden. The strongly coupled duals of type IIA and Type IIB orientifoldsare given by singular M-theory compactifications on G2 manifolds and, respectively,by F-theory compactifications on elliptically fibred Calabi-Yau fourfolds [12]. Thelocal model building rules for such F-theory compactifications have been worked outrecently in [13, 14, 15,16, 17,18, 19]; For recent studies of 7-branes from the F-theory

perspective see [20, 21,22, 23, 24].

On the other hand, investigations of non-perturbative corrections for Type II

orientifold models [25, 26,27] have revealed that the 10105H Yukawa coupling canbe generated by Euclidean D-brane instantons wrapping suitable cycles in theinternal manifold with the right zero mode structure [28]. Of course these couplings

are suppressed1 by the exponential of the instanton action (Tinst) = g1s Vol(). It iscrucial to appreciate that this suppression is not tied to the inverse gauge coupling ofthe Standard Model, as would be the case for effects related to gauge (as opposed tostringy) instantons, but can in principle take any value, depending on the geometricdetails of our compactification manifold. This feature, which holds both for TypeIIA and Type IIB orientifolds, opens up the prospect of SU(5) GUT model buildingalready in the limit Tinst 0 of perturbative Type II orientifolds. Once we alsotake the nice features of moduli stabilisation in Type IIB into account, one mightseriously hope that the strong coupling limit of Type IIB orientifolds, either in theirgenuinely F-theoretic disguise or in their perturbative description as D-brane modelswith O7- and O3- planes and Tinst 0, may indeed provide a promising startingpoint to construct realistic GUT models.

In the recent work [14,16], the authors draw the first conclusion (see also [13,17]).Taking into account that F-theory models on elliptically fibred fourfolds can admitdegenerations of the elliptic fibre such that exceptional gauge groups appear natu-rally, these references pursue the program of studying GUT type F-theory compacti-fications. As a physical input, the authors of [14, 16] propose the working hypothesis

1Note that for a Georgi-Glashow SU(5) GUT the 10105H Yukawa gives masses to up-typequarks, whereas for flipped SU(5) it provides the down-type quark masses. In the latter case, theexponential suppression might explain the little hierarchy between up and down quarks [28].

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that the Planck scale ought to be decouple-able from the GUT scale, even if only inprinciple. From the Type IIB perspective, these F-theoretic models do not only con-tain usual D-branes, but also so-called (p,q) seven-branes which carry charge under

both the R-R and the NS-NS eight-forms. The new non-perturbative states, such asthe gauge bosons of exceptional groups or the spinor representations of SO(10), aregiven by (p,q) string junctions starting and ending on these branes. Unlike funda-mental strings, these string junctions can have more than two ends thus providingextra states.

From the guiding principle of decoupled gravity it is further argued in [16] that

the (p,q) 7-branes should wrap shrinkable four-cycles in the internal geometry. Theseare given by del Pezzo surfaces2. Lacking a global description of Calabi-Yau fourfoldswith the desirable degeneration, the authors provide a local set-up of singularities or(p,q) 7-branes and line bundles so that the GUT particle spectrum is realised. At

a technical level there arises a challenge with GUT symmetry breaking because atheory on a del Pezzo surface has neither adjoints ofSU(5) nor discrete Wilson linesat its disposal to break SU(5) to SU(3) SU(2) U(1)Y . One option would be toadopt the philosophy of heterotic compactifications and embed a further non-trivialU(1) line bundle, as discussed for the heterotic string originally in [31] and morerecently in [32,33,34,35]. For line bundles non-trivial on the Calabi-Yau manifold, theassociated U(1) generically becomes massive due to the Stuckelberg mechanism, butin presence of several line bundles special linear U(1) combinations remain massless.In the heterotic context of [34], in order to maintain gauge coupling unificationwithout relying on large threshold corrections it is necessary to consider the large gslimit of heterotic M-theory [36].

As a new and very central ingredient the authors of [16] propose to break the GUTgauge group instead by a line bundle embedded into U(1)Y such that it circumvents

the sort of no-go theorem mentioned above (see also [17]). The idea is to support thebundle on a non-trivial two-cycle inside the del Pezzo surface which is trivial on theambient four-fold base. It was argued in [16] that with this mechanism some of thenotorious problems of GUTs such as the doublet-triplet splitting problem, dangerousdimension five proton decay operators and even neutrino masses can be addressedand actually solved by appropriate choices of matter localisations and line bundleson the del Pezzo divisors. Studies of supersymmetry breaking mechanisms for this

class of local models have appeared in [37, 38,39].

The Planck-scale decoupling principle might be a justification for a local approach

2Local quiver type models on del Pezzo singularities have been studied, for example, in [29, 30].

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to string model building (and indeed a quite constraining one), but in absence of arealisation of the described mechanisms in globally consistent string compactifica-tions it remains an open question if these local GUT models do really consistently

couple to gravity. In fact, it is the global consistency conditions of string theorywhich decide whether a given construction is actually part of the string landscape ormerely of the swampland of gauge theories. At a technical level, it is therefore nowonder that they constitute some of the biggest challenges in string model building,and many interesting local constructions fail to possess a compact embedding sat-isfying each of these stringy consistency conditions. For example, whether or not agiven U(1)Y flux actually leads to a massless hypercharge depends on the global em-bedding of the divisor supporting the 7-brane into the ambient geometry and cannot

be decided within a local context.

In F-theory the global consistency conditions, in particular the D7-brane tad-

pole cancellation condition, are geometrised: They are contained in the statementthat indeed a compact elliptically fibred fourfold exists such that the degenerationsof the fibre realise the GUT model. This is a very top-down condition and given

the complexity and sheer number of fourfolds it is extremely hard to implement inpractise.

It is the aim of this paper to address these global consistency conditions bytaking a different route. As described above Georgi-Glashow SU(5) GUT modelscan naturally be realised on two-stacks of D7-branes in a perturbative Type IIBorientifold. Here we have quite good control over the global consistency conditionsas they are very similar to the well-studied Type I or Type IIA orientifolds. Therefore,

our approach is to first construct a GUT model on a Type IIB orientifold, satisfy allconsistency conditions, check whether the top quark Yukawa is really generated byan appropriate D3-instanton and then take the local Tinst. 0 limit3.

To follow this path, we start by partly newly deriving, partly summarising themodel building rules for Type IIB orientifolds. We then study how much of theappealing structure proposed in the F-theoretic context, such as the U(1)Y GUTgauge breaking, can already be realised in perturbative Type IIB orientifolds onCalabi-Yau threefolds with intersecting D7-branes wrapping holomorphic surfaceswith non-trivial vector- or line-bundles.

In order for the U(N) gauge factors on the D7-banes not to exhibit chiral mul-

3Here we make the working assumption that a Type IIB vacuum satisfying all K-theory andsupersymmetry constraints has an uplift to F-theory on a Calabi-Yau fourfold. We are aware thatthis may not be so straightforward to show [40].

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manifolds naturally containing del Pezzo surfaces. These compact manifolds haverecently been considered in [41] and contain the kind of holomorphic surfaces allow-ing for the realisation of many of the GUT features we are interested in. They can

be described as elliptic fibrations over del Pezzo surfaces dPn, n = 1, . . . , 8 and theirvarious connected phases related via flop transitions. In the elliptic fibration itself,besides the dPn basis, we find various dP9 surfaces, which via the flop transitions be-come dP8 surfaces or P2 surfaces with more than nine points blown up. In the courseof this section, to define the orientifold actions we have to investigate the existence ofappropriate involutions. In order not to interrupt the physics elaborations too muchthis rather technical though central discussion has mainly been shifted to appendixA. The mathematically interested reader is encouraged to consult this appendix for

more details on the classification of involutions and the determination of the fixedpoint loci. As an important part of our analysis we will prove the swiss cheesestructure of those del Pezzo transitions where the dP9 surfaces have all been floppedto dP8 surfaces. We will show that as a consequence of this structure the D-termsupersymmetry conditions force the cycles supporting D-branes to take a vanishingvolume, that is, they are dynamically driven to the quiver locus.

In Sections 5 and 6 we present some first concrete SU(5) GUT models. These arethe outcome of an essentially manual search which has succeeded in implementingall known global consistency conditions. As a warm-up, Section 5 discusses at lengthan SU(5) model on the Weierstra model over dP2 with two chiral families of SU(5)GUT matter, one vector-like pair of Standard Model Higgs and no chiral exotics.The GUT matter transforming in the 10 is localised in the bulk of the GUT branes,while the 5 and the Higgs pair arise at matter curves. Upon breaking SU(5) bymeans of U(1)Y flux there arise extra vector-like pairs of Standard Model matter.As one of its phenomenologically appealing features, this model contains a 10105HYukawa coupling of order one induced by a Euclidean D3-brane instanton in the limitTinst. 0, but the global consistency conditions do not allow for the constructionof a three-generation model on this particular geometry. To remedy this we present,in Section 6, a string vacuum of a similar type on the del Pezzo transition of thisWeierstra model, but featuring three chiral families of Standard Model matter, no

chiral exotics and only two pairs of extra vector-like states. As a consequence ofthe swiss cheese structure of the manifold, the D-term supersymmetry conditionsdrive the vacuum to the boundary of the Kahler cone. This can be avoided in a

three-generation GUT model on the Weierstra model over dP3 as discussed in theremainder of this section. The key phenomenological achievements and drawbacksof these three examples are summarised in Tables 9, 13, and 15.

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fold and the cycles wrapped by the D-branes. The involution splits the cohomologygroups into eigenspaces, and allows one to identify the spectrum preserved by theorientifold. Focusing on the bulk fields corresponding to closed string excitations,

one notes that Hp,q

(Y,C) splits as H

p,q

+ Hp,q

with dimensions hp,q

respectively. Onethus obtains the complex dilaton = C0 + ie, h1,1+ complexified Kahler moduli TI

and h1,1 B-field moduli Gi given by [49, 50]

TI =

+I

, Gi =

i

, = eB

eRe

eiJ

+ iCRR

(1)

where CRR = C0 + C2 + C4. The cycles +I and

i form a basis of the homology

groups H+2,2 and H1,1, respectively. We will call the continuous moduli G

i simply Bmoduli, since they encode the variations of the NS-NS and R-R two-forms. While no

dynamical moduli are associated with the reduction of the B-field along the positive

2-cycles I+ H+1,1 there can still be discrete non-zero B-flux 12 I+ B = 12 . Thissurvives the orientifold action due to the axionic shift symmetry

I+

B I+

B + 2

and will sometimes be referred to as B+ flux. In the following we will determinewhich quantities in the four-dimensional action depend on which of these closedstring moduli.

2.1 Intersecting D7-Branes With Gauge Bundles

We first discuss the inclusion of space-time filling D7-branes in more detail. Considerwrapping a stack ofNa D7-branes around a four-cycle Da in Y. The calibration con-dition for the D7-branes requires Da to be a holomorphic divisor [51]. The orientifoldsymmetry maps Da to its orientifold image Da so that in the upstairs geometryeach brane is accompanied by its image brane. There are three different cases to bedistinguished:

1. [Da] = [Da],2. [Da] = [Da] but Da = Da point-wise, and3. Da = Da point-wise, that is, D7-branes coincide with an O-plane.

In the first two situations, the D7-brane may or may not intersect an O7-plane. Forvanishing gauge flux, branes of the first type carry unitary gauge groups, while thoseof the other two types yield symplectic or orthogonal gauge groups. In absence of

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CFT methods to uniquely distinguish SO vs. SP Chan-Paton factors the rule ofthumb is that a stack of Na branes plus their Na image branes on top of an O7/+-plane6 gives rise to a gauge group SO(2Na)/SP(2Na). The same configuration along

a cycle of type 2 with locally four Dirichlet-Neumann boundary conditions to theO7/+-plane yields gauge group SP(2Na)/SO(2Na).

Gauge fluxes on D7-branes

Each stack of D7-branes can carry non-vanishing background flux for the Yang-Mills field strength Fa. Recall that the field strength Fa appears in the Chern-Simons and DBI action only in the gauge invariant combination Fa = Fa + B1,where : Da Y denotes the embedding of the divisor Da into the ambient space.Therefore all physical quantities depend a priori only on Fa. However, as we willdescribe in detail below, with the exception of the D-term supersymmetry conditiononly the discrete B+-flux effectively enters the consistency conditions.

A consistent configuration of internal gauge flux is mathematically described interms of a stable holomorphic vector bundle7 by identifying the curvature of itsconnection with the Yang-Mills field strength. For all concrete applications in thisarticle it will be sufficient to restrict ourselves to the simplest case of line bundlesLa, corresponding to Abelian gauge flux. For a single D-brane wrapping a simplyconnected divisor these are determined uniquely by their first Chern class c1(La)

as an element of H2(Da) or equivalently by a two-cycle la with class in H2(Da) asLa = O(la). For stacks of several coincident branes wrapping the divisor Da we alsohave to specify the embedding of the U(1) structure group of the line bundle intothe original gauge group on the branes.

Let us start with a stack of Na branes of type 1 and decompose the background

value of the physical Yang-Mills field strength FasFa = T0 (F(0)a + B) +

i

TiF(i)a . (2)

Here T0 = 1NaNa refers to the diagonal U(1)a U(Na) while Ti are the tracelessAbelian8 elements of SU(Na). This defines the line bundles L

(i)a as

c1(L(0)a ) =

1

2(F(0)a +

B) H2(Da), c1(L(i)a ) =1

2F(i)a H2(Da). (3)

6A O7/+-plane carries 8/ + 8 times the charge of a D7-brane in the upstairs geometry.7More generally, gauge flux is described by coherent sheaves, but for our purposes it suffices to

consider vector bundles on the divisors.8This can be generalised to non-Abelian vector bundles. For example, on a stack of Na coincident

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Note that in view of the appearance of Fa in all physical equations the B-field is tobe included in c1(L0a).

While the effect of L(0)a is merely to split U(Na) SU(Na) U(1)a, the other

L(i)a will break SU(Na) further. The relevant example we will be studying in detail isthe breaking ofU(5)a SU(5)U(1)a SU(3)SU(2)U(1)YU(1)a by meansof diagonal flux and another line bundle corresponding to the hypercharge generatorTY. Note that the Abelian gauge factors may become massive via the Stuckelbergmechanism [54].

For a stack of 2Na invariant branes of type 2 and 3 a non-trivial bundle L(0)a

breaks SO(2Na)/SP(2Na) SU(Na) U(1)a and the embedding of L(i)a works inan analogous manner. We will be more specific in the context of the concrete setupdescribed in Subsection 3.1.

In general it is possible for some components of9 c1(La) along H2(Da) to be trivial

as elements of H2(Y). Recall that the inclusion : Da Y defines the pushforward : H2(Da) H2(Y) and pullback : H2(Y) H2(Da). Then one can split La as

La = La Ra, (4)

with La = O(a) defined as a line bundle on the Calabi-Yau Y. The part ofLa trivialin Y, denoted as Ra = O(ra), corresponds to a two-cycle ra which is non-trivial onDa but a boundary in Y, that is, [ra] ker(). The possibility of considering suchgauge flux in the relative cohomology ofDa in Y was first pointed out in [55, 56] andits relevance for model building was stressed in [30, 41].

We need to understand which quantities are affected by a relative flux R. In thiscontext, we will make heavy use of the following integrals

Da

c1(Lb) c1(Lc) =

Y

[Da] c1(O(b)) c1(O(c)) = abc,Da

c1(Lb) c1(Rc) =

Dab

c1(Rc) = 0, (5)Da

c1(Rb) c1(Rc) = abc ,

branes one can define a holomorphic rank na bundle (with na dividing Na) and embed its structure

group U(na) into the original U(Na) theory. This breaks the four-dimensional gauge group downto the commutant U(Na/na) of U(na) in U(Na). See [52,53] for a general discussion in terms ofD9-branes on Calabi-Yau manifolds.

9In the sequel we will sometimes omit the superscripts in L(i)a to avoid cluttering of notation.

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where abc and abc are not necessarily zero. We conclude that the integral over adivisor of a pull-back two form wedge a two-form which is trivial in Y vanishes. Aswill be detailed below, this implies that a bundle in the cohomology which is trivial

in Y but non-trivial on Da does not affect the chiral spectrum, the D-term supersym-metry conditions and the D5-brane tadpole of the brane configuration. However, itdoes affect the gauge symmetry, the D3-brane tadpole and the non-chiral spectrumof the model.

Quantisation condition

Essential both for consistency of the theory and for concrete applications is to ap-preciate the correct quantisation conditions on the gauge flux. Following [42] theyare determined by requiring that the worldsheet path integral for an open stringwrapping the two-surface with boundary along Da be single-valued. Considerfirst a single brane wrapping the divisor Da and carrying Abelian gauge flux Fa. Thequantity to be well-defined is given by

Pfaff(D) exp(i

Aa) exp(i

B) (6)

in terms of the Pfaffian of the Dirac operator, the connection A of the Abelian gaugebundle and the B-field. IfDa is not Spin, that is, ifc1(KDa) = 0 mod 2, the Pfaffianpicks up a holonomy upon transporting around a loop on Da [42]. This holonomymust be cancelled by the second factor in eq. (6). For internal line bundles this isguaranteed if the gauge flux obeys the condition

Fa +1

2

KDa Z H2(Da,Z), (7)

or equivalently, using our convention eq. (3),

c1(La) B + 12

c1(KDa) H2(Da,Z). (8)

Note in particular that for trivial B flux along Da, B = 0, the Abelian gaugebundle on the single brane Da has to be half-integer10 quantised if the divisor Da is

not Spin.10The quantisation condition eq. (8) with non-trivial B-field is related to the concept of vector

bundles without vector structure [57] in Type I theory as studied recently, for example, in [58, 59].

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This condition is readily generalised to line bundles on stacks of D-branes. Theprobe argument of [42] now implies that the path integral has to be well-defined forevery disk worldsheet with boundary on each of the branes in the stack of Na

D-branes wrapping Da. This requires

T0 (c1(L(0)a ) B) +

i

Ti c1(L(i)a ) +

1

2T0 c1(KDa) H2(Da,Z)NaNa . (9)

where the notation on the right hand side means that all elements of the Na Namatrix on the left hand side are in H2(Da,Z). One concludes that depending on the

precise from of Ti the bundles L(i)a can in general be fractionally quantised, a fact

that will be very important for our applications.

A second constraint arises for the continuous B moduli in H2: the restrictionto Da of the characteristic class of the B-field, introduced in [42], has to equal the

third Stiefel-Whitney class ofDa. Recall from [42] that modulo torsion, is given bythe field strength H = dB and that for complex divisors the third Stiefel-Whitneyclass is always zero. Moreover, for all surfaces considered in this paper, we haveH3(Da,Z) = 0 so that H = dB always restricts to zero on the divisor. Thereforeno further condition on the B-field moduli Gi introduced in (1) arises from theseconsiderations.

Orientifold action

Let us now describe the orientifold action on the gauge flux. To this end note thatthe orientifold action : D

D induces a map on cohomology, : H2(Da,Z)

H2(Da,Z). The full orientifold action on a vector bundle on Da is given by thecomposition p. Here p acts as dualisation, La La . In particular, the Cherncharacter of the image bundle is

chk(La) = (1)kchk(La) = chk(La ). (10)

We now discuss the three cases introduced at the beginning of Subsection 2.1in turn. In the first situation, where not even the homology class of the brane ispreserved, one can define two divisors Da and two vector bundles L

a by setting

Da = Da (Da) , La |Da = La , La |Da = La, (11)where Da is the cycle Da with reversed orientation. Upon setting H2(D+a ) =H2(Da) H2(Da) and decomposing the latter into positive and negative eigenspacesunder , H2(D+a ) = H

2+(D

+a ) H2(D+a ) [55], it follows that c1(L+a ) H2(D+a ).

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In the second case the homology class is preserved but the brane is not point-wisefixed. Hence, the homology class of Da in eq. (11) is trivial and we can use [Da] =12

[D+a ]. The degree-2 cohomology group of Da thus splits again as H2(Da,Z) =

H2

+(Da) H2

(Da). On the covering space of the orientifold one requires an evennumber of branes in the homology class ofDa which are pairwise identified under theinvolution . Clearly, this corresponds to an integer number of branes on D+a . TheChern class c1(La) on Da is in the full H2(Da) and (Da, La) is mapped to (Da, La)as in eq. (10).

In the third case, for Da on top of the orientifold, H2(Da) = H

2+(Da) and (Da, La)

is mapped to (Da, La ). This case directly parallels the situation for D9-branes inType I compactifications. An odd number of branes stuck on top of the orientifoldplane is not possible, as discussed recently in [23]. Formally we therefore workupstairs with the system 2Na Da carrying the invariant bundle La La .

2.2 Tadpole Cancellation for Intersecting D7-Branes

In consistent compactifications it is crucial to cancel the tadpoles of the space-timefilling intersecting D7-branes. Satisfying the tadpole cancellation condition ensuresthat the spectrum is free of non-Abelian gauge anomalies. In general, D7-branes carryalso induced D3- and D5- charges arising due to a non-trivial gauge-field configurationon the seven branes and through curvature corrections. All induced tadpoles for acompactification have to be cancelled.

Throughout this article we will be working upstairs on the ambient Calabi-Yau

manifold before taking the quotient by . Recall that the K-theoretic charges ofa D7-brane and the O7-plane along divisors Da and DO7 are encoded in the Chern-Simons coupling to the closed RR-forms 2

R1,3Da

2p C2p. Concretely these are

given by

SD7 = 2

R1,3Da

2p

C2p tr

e12Fa A(T D)

A(ND),

SO7 = 16R1,3DO7

2p

C2p

L( 14 T DO7)

L( 14 NDO7)(12)

in terms of the A-roof and the Hirzebruch genus associated with the tangent andnormal bundles to the respective divisors. The D7-, D5-, and D3-brane chargesfollow upon straightforward decomposition of eq. (12).

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form flux, which is in particular important for complex structure moduli stabilisation.The induced D3-charge on the O7-planes is given by

QO7 =

(DO7)

6 =

1

6 Y c2(DO7) [DO7]=

1

6

Y

[DO7]3 + c2(TY) [DO7] . (18)

If a stack of Na D7-branes wraps a smooth divisor of type 1 or 3, as defined onpage 11, their D3-charge reads

QaD7 = Na(Da)

24+

1

82

Da

tr F2a (19)

with

1

82trF2a = 12

I,J

tr[TITJ] c1

L(I)a

c1

L(J)a

. (20)

More subtle is the case 2, since eq. (17) will be modified as discussed in [21, 23]. Onereplaces the Euler characteristic by

QaD7 = Nao(Da)

24+

1

82

Da

tr F2a , o(Da) = (a) npp, (21)

where a is an auxiliary surfaces a obtained by blowing up the singular points inDa, while npp counts the number of pinch points in Da.

The relation to the F-theory D3-brane tadpole condition becomes obvious if oneslightly reorders the terms in eq. (17) and divides by two,

ND3 +Nflux

2+ Ngauge =

NO34

+(DO7)

12+a

Nao(Da)

24(22)

with

Ngauge = a

1

82

Da

tr F2a = 1

2

a

Na

Da

I,J

tr[TITJ] c1

L(I)a

c1

L(J)a

. (23)

The right-hand side of equation (22) is precisely (Y4)/24 in the F-theory lift of thisType IIB orientifold, where Y4 denotes the Calabi-Yau fourfold. This implies thatgenerically each topologically different arrangement of D7-branes cancelling the RR

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eight-form tadpole constraints lifts to a different Calabi-Yau fourfold with differentEuler characteristic. For the trivial solution with eight D7-branes placed right on topof the smooth orientifold plane with npp = 0 the right hand side of (22) simplifies toNO3

4 +(DO7)

4 . It is a consistency check that this number is indeed an integer.Let us emphasise that for the cancellation of anomalies only the D7 and D5-

tadpole constraints are important. The D3-brane tadpole is in some sense onlyrelated to the non-chiral sector of the D-brane theory. This is related to the factthat a D3-brane can never carry any chiral modes, as it can in principle be moved toa position away from the D7-branes. The expectation is that a globally consistentType IIB orientifold model with a supersymmetric D7- and D5-brane sector lifts upto F-theory on a compact Calabi-Yau fourfold. The cancellation of the D3-brane

tadpole is an additional attribute both in Type IIB orientfolds and in F-theorymodels. Taking also into account that for moduli stabilisation and the realisation

of inflation, the presence of (a small number of) anti-D3-branes is very welcome, inthis paper we take all the D7- and D5-brane supersymmetry constraints very seriousbut are a bit more relaxed about the existence of anti- D3-branes in the system. Infact, we will find that in our semi-realistic GUT examples the D3-brane tadpole caneasily be saturated by already modest addition of gauge fluxes on the D7-branes.

Role of B-moduli

Before proceeding we would like to comment on the role of the continuous B-moduliappearing in F and thus in the D-brane charges. It is natural to wonder how toreconcile their contribution with the discrete nature of a quantity such as the D5- orD3-brane charge.

In fact the B moduli decouple from the tadpole equations by means of the D7-brane tadpole cancellation condition (13) and the simple observation that the Bfield restricts trivially to the O7-plane,

DO7

B = 0 , H2(Y) . (24)

Concretely, the B-contribution to the D5-brane tadpole condition (16)

a

Na Y [Da] B + [Da] B = 8Y [DO7] B = 0 (25)indeed vanishes due to (13) and (24).

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To isolate the B-moduli in the D3-brane tadpole let us introduce the quantity

c1(

La) = c1(La) B (26)

and rewrite the induced D3-brane tadpole as

a

Na

Da

c1(La) + B2 +

Da

c1(La) + B2 . (27)

For simplicity we are sticking to gauge flux 12F= T0c1(La). For the mixed term wefind that

a

Na

Y

[Da] c1(La) + [Da] c1(La) B = 0 , (28)

where we have used the D5-brane tadpole cancellation condition (16). Finally, weevaluate

a

Na Y

[Da] B2 + [Da] B2 = 8 DO7

B2 = 0 (29)

so that as anticipated the continuous B-moduli do not appear in the tadpole can-cellation conditions.

K-Theory charge cancellation

Apart from cancellation of these homological charges, also all K-theoretic torsion

charges have to sum up to zero. In general it is a very non-trivial task to computeall in particular torsional K-theory constraints. However, according to the probe

brane argument of [60] cancellation of torsion charges is equivalent to absence of aglobal SU(2) Witten anomaly on the worldvolume of every probe brane supportingsymplectic Chan-Paton factors. In concrete compactifications this amounts to re-quiring an even number of fundamental representations in the sector between thephysical D7-branes and each symplectic probe brane. Note that in a concrete model

determining all symplectic four-cycles is also far from trivial.

2.3 The Massless Spectrum

For applications to phenomenology it is essential to understand the massless matter

arising from open strings stretching between two stacks of D7-branes.Non-chiral matter transforming in the adjoint representation emerges from strings

with both endpoints on the same D-brane along Da. It consists of the vector multiplet

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together with h1,0(Da) and h2,0(Da) chiral multiplets describing the Wilson line and

deformation moduli of the D7-branes. Matter in the bifundamental representation12

(Na, Nb) and (Na, Nb) arises from open strings stretching between two different D7-

branes in the (a, b) and (a, b) sector, respectively. Intersections between a brane andits image, that is, of type (a, a), yield matter in the (anti)symmetric representation.For example, if all branes are on top of a O7()-plane, then all states in the (a, a)sector are anti-symmetrised. On an invariant brane with four Dirichlet-Neumannboundary conditions with an O7()-plane, the (a, a) states are symmetrised. Thechiral spectrum is summarised in Table 1, see also [61]. For the concrete computation

sector U(Na) U(Nb) chirality

(ab) (1) (1) Iab

(ab) (1) (1) Iab

(aa) (2) 1 12 (Iaa + 2IO7a)

(aa) (2) 1 12 (Iaa 2IO7a)

Table 1: Chiral spectrum for intersecting D7-branes. The subscripts denoteU(1) charges.

of the chiral index Iab and to determine the vector-like spectrum we have to distin-guish two situations according to the localisation of matter on sub-loci of different

dimensions. For simplicity we again only discuss the case where all D7-branes carryholomorphic line bundles.

2.3.1 Matter Divisors

For two D7-branes wrapping the same divisor Da = Db = D and carrying linebundles La and Lb, matter in the bifundamental representation (Na, Nb) is countedby the extension groups [62]

Extn(La, Lb), n = 0, . . . 3, (30)

12For the general overview we only consider diagonal embeddings and postpone a discussion ofmore general line bundles to the applications in Section 5.

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where i : D Y defines the embedding of D in the Calabi-Yau Y. The value n = 1refers to anti-chiral multiplets transforming as (Na, Nb), while n = 2 correspondsto chiral multiplets in the same representation. For consistency, the states counted

by the groups corresponding to n = 0 and n = 3 have to absent. These statesdo not correspond to matter fields but rather gauge fields and have been namedghosts in [13]. We show in Subsection 2.4 that for supersymmetric configurationswith the Kahler form inside the Kahler cone these ghosts are automatically absent.By running through the spectral sequence, one can show that the sheaf extensiongroups eq. (30) are given by appropriate cohomology groups for line bundles on thedivisor D, concretely

Ext0(La, Lb) = H0(D, La Lb ),Ext1(La, Lb) = H1(D, La Lb ) + H0(D, La Lb ND),

Ext

2

(La, Lb) = H2

(D, La Lb ) + H1

(D, La Lb ND),Ext3(La, Lb) = H2(D, La Lb ND). (31)

By Serre duality and ND = KD we can relate Hi(D, La Lb ND) = H2i(D, La Lb). It straightforwardly follows that for the chiral index Iab counting bifundamentalmatter transforming as (Na, Nb) one obtains

Ibulkab =3n=0

(1)n dim Extn(La, Lb)

= (D, La Lb ) (D, La Lb ND)=

Y[D] [D] ( c1(La) c1(Lb) ) .

(32)

In these conventions Iab > 0 if there is an excess of chiral states in the representation

(Na, Nb). Note that this expression only depends on the components ofc1(Li) whichare non-trivial on the ambient Calabi-Yau manifold, cf. eq. (5).

We have the additional freedom to twist the line bundle La on D by a line bundleRa with Ra = O. This does not change the chiral spectrum, though it can changethe vector-like one and will in general contribute to the D3-tadpole.

2.3.2 Matter CurvesIf the two D7-branes wrap different divisors Da and Db intersecting over a curve Cof genus g and carrying line bundles La and Lb, the cohomology groups counting

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symmetry with Killing vector XLa is of the form Da = XLa MLK, where K is the

Kahler potential. Let us recall the induced gauging for the complex scalars TI andGi defined in (1). A gauging ofTI can be induced for a non-trivial line bundle on a D7-

brane (Da, La), while Gi

can be gauged if there exists a Da which is not homologousto Da, that is, if we are in the case 1 defined at the beginning of Subsection 2.1,page 11. The Killing vectors for these gaugings are of the form

Xa I =

Da

[+I ] c1(La) , Xia = Da

[i] B , (37)

where +I , i are the four- and two-cycles introduced in eq. (1) to define TI, G

i. Notethat there is no continuous moduli dependence in Xa I since we have explicitly splitoff the B field as in eq. (26). One next notes that [49, 63]

TIK rI , GiK si, (38)

where rI, si arise in the expansions J = rI [+I ] and JB = si[i]. It is important tonote that the expression for TIK in eq. (38) is also valid away from the large volumelimit. For example, one of the rI can become small while GiKwill receive additionalcorrections, for example, due to world-sheet instantons. We thus encounter a modulidependent Fayet-Iliopoulos term for the configuration of the form [51, 55]

a Da

J (c1(La) + B) = Da

J c1(La) . (39)

Note that a depends on the pullback of the Kahler form J ofY to the D7-brane andas a consequence of eq. (5) only on the components ofc1(La) which are non-trivial onY. Furthermore the B

-moduli, encoded in c1(La), do not drop out of the D-terms.

For vanishing VEVs of the chiral fields charged under the U(1) supported on the D7-branes, the D-term supersymmetry condition requires these FI-terms to vanish. Thisimposes conditions on the combined Kahler and B moduli space. As long as theKahler moduli are chosen such that J is indeed invariant under the orientifold action,

the Fayet-Iliopoulos term for (Da, La) and (Da, La) coincide. We will encounter that

in a concrete example in Subsection 6.2.

For line bundles La = O satisfying the D-term constraint eq. (39), we will nowderive two important consequences:

No ghostsFirst, we realise that the FI-term is nothing else than the slope (La) of the linebundle La. Now to come back to the question of ghosts in the massless spectrum in

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eq. (31), it is important to recall the general fact that

For two vector bundles Va, Vb of equal slope and rank, (Va) = (Vb) andrk(V

a) = rk(V

b), the existence of a map 0

Va

Vb

implies that Va

= Vb.

We thus conclude for the extension groups between two supersymmetric line bundlesLa = Lb that Ext0(La, Lb) = Ext3(La, Lb) = 0. Indeed, if H0(D, La Lb )were non-vanishing, we could define a map 0 O La Lb where both O and,by hypothesis, La Lb have vanishing slope. Therefore, La = Lb in contradictionto our assumption. The same reasoning for the dual bundle La Lb shows thatExt3(La, Lb) = 0.

D3-tadpole contribution

Second we note that for line bundles with vanishing slope for a Kahler form insidethe Kahler cone, the contribution of the gauge flux to the D3-brane tadpole alwayshas the same sign

Na2

Da

c21(La) 0. (40)

Indeed, on a surface Da the set of c1(La) with vanishing slope is given by H2(Da) {M M}, where M denotes the Mori cone ofDa. However, the Mori cone containsall classes C with C2 > 0 and C K > 0. Therefore, c21(La) 0. This result impliesthat for supersymmetric brane configurations the possible choices of line bundles arerather limited if we do not want to introduce antiD3-branes in the system.

Finally, let us mention that the other supersymmetry conditions, namely theholomorphy of the divisor and the bundle, arise from a superpotential

WD7 =

C(La,La)

, (41)

where C(La, La) is a chain ending on the two-cycle Poincare dual to c1(L+a ) on thedivisor Da + Da.

3 SU(5) GUTs and Their Breaking

After this discussion of the general model building rules for Type IIB orientifoldswith O7- and O3- planes we can now become more specific about the realisation of

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SU(5) Georgi-Glashow GUTs. Parts of the logic are very similar to the implemen-tation of GUT models in Type IIA intersecting D-branes [43, 11, 45] as described, forexample, in [11, 28]. Let us first transfer these rules to our Type IIB setting. Then

we move forward to describe how the mechanism of GUT symmetry breaking viaU(1)Y flux, exploited by [16] in the local F-theory context, can also be realised inthis perturbative orientifold limit.

3.1 Georgi-Glashow SU(5) GUT

The starting point is the construction of a U(5) U(1) gauge theory from a stackof five D7-branes wrapping a four-cycle Da and one additional brane wrapping Db.These brane stacks carry holomorphic bundles La and Lb, respectively.

The orientifold action maps (Da, La) (Da, La) (and similarly for (Db, Lb)). Aspreviously discussed, this includes the case that Da is invariant under . First wediagonally embed two line bundles La and Lb by identifying their structure groupwith the diagonal U(1)a and U(1)b, respectively. Each of the two Abelian factorsU(1)a and U(1)b separately acquires a mass by the Stuckelberg mechanism as long

as Da and Db are non-trivial homology cycles [54].

A more group theoretic way of describing the gauge group and its matter contentis to start with an SO(12) gauge group. The embedding of two line bundles with

structure groups U(1)a,b can break this to U(5) U(1), where the generators of thetwo U(1)s are embedded into SO(12) as

U(1)a diag(155, 0 | 0, 155 ) ,U(1)b diag(055, 1 | 1, 055 ) . (42)

The adjoint of SO(12) decomposes into SU(5) U(1)a U(1)b representations as

[66] = [24](0,0) + [1](0,0) + [10](2,0) + [10](2,0) + [5](1,1) + [5](1,1)+[5](1,1) + [5](1,1). (43)

To ensure absence of a massless combination of U(1) factors we require that [Da]and [Db] be linearly independent in H4(Y, Z). Note that in the presence of furthertadpole cancelling D7-branes it has to be ensured that the full mass matrix is of

maximal rank.

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The MSSM spectrum

The massless states transforming in the adjoint representation are given by the de-formations of the four-cycles, which are counted by H1(D,

O) (Wilson lines) and

H2(D, O) (transversal deformations). In principle we could allow for precisely onesuch adjoint of SU(5), which might break the SU(5) symmetry to the Standardmodel by the Higgs mechanism. An example of such a surface with h(2,0) = 1 is K3.However, a complete GUT model relying on this mechanism would have to addressthe generation of a suitable potential for the GUT Higgs field from string dynamics

such that SU(5) is broken dynamically to the Standard Model. Since we will ratherbe breaking the GUT symmetry by embedding U(1)Y flux, we insist that the SU(5)stack wraps a rigid four-cycle. This is satisfied for del Pezzo surfaces, which haveh1,0(D) = h2,0(D) = 0. In view of the rules of Table 1 the charged GUT spectrum

requires the chiral intersection pattern listed in Table 2.

state number sector U(5) U(1)

10 3 (aa) (2) 1

5 3 (ab) (1) (1)

1N 3 (bb) 1 (2)

5H+ 5H 1 + 1 (ab) (1) (1)

Table 2: Chiral spectrum for intersecting D7-brane model. The indices denote

the U(1) charges. The last line gives the Higgs particles.

The first two lines contain the antisymmetric representation 10 ofSU(5) and thefundamental 5. The states from the bb sector are necessary to satisfy the formalU(Nb) anomaly (3 (4 + 1) 3 5 = 0) and carry the charges of right-handedneutrinos. States from the (ab) carry the right quantum numbers to be identifiedwith the pair of Higgs fields 5H + 5H. However, one can also realise the Higgs fieldsfrom intersections (ac) between the SU(5) brane stack and a third one. In contrast

to SO(10) GUTs, here all massless fields are perturbatively realised by open string

stretched between stacks of D7-branes.The various fields are localised on the intersection of the various divisors. As

mentioned already in Subsection 2.1, these are either curves or divisors. In the latter

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case one has to compute cohomology classes over a del Pezzo surface, which in generalgives also vector-like matter. On the contrary, if two divisors intersect over a curvevector-like states are much easier to suppress. We will exemplify this feature in the

concrete examples to be discussed later.

Yukawa couplings

The Yukawa couplings which give masses to the MSSM fields after GUT and elec-

troweak symmetry breaking are

10(2,0) 10(2,0) 5H(1,1), 10(2,0) 5

(1,1)5H

(1,1)1

(0,2)N 5

(1,1)5H

(1,1) , (44)

where the upper indices denote the Abelian U(1)a U(1)b charges. If as indicatedwe realise the matter and Higgs fields as in Table 2, the last two Yukawa couplings,

i.e. the ones generating masses for the d-quarks and leptons, are allowed already atthe perturbative level. For them to be present the wave functions of the masslessmodes have to overlap. If all states are localised on curves, this means that the threedivisors have to meet at a point. On a Calabi-Yau threefold, this is generically thecase. Note that to first order the wavefunctions of the fields localise strictly alongthe matter curves and these perturbative Yukawa couplings are of rank one. Only

higher order corrections to the wavefunction profile are responsible for a non-trivialfamily structure. If on the other hand the 10 and the 1N arise from the bulk of theGUT brane and the U(1)b brane while the 5 and the Higgs are localised on curves,the perturbative Yukawa couplings involve the triple-product of the restriction of

corresponding powers of La and Lb to the matter curve.By contrast, it is obvious from their U(1) charges in eq. (44) that the u-quark

Yukawa couplings 10105H are perturbatively forbidden. For quite some time thiswas considered the main obstacle to the construction of open string SU(5) GUTmodels. This no-go was bypassed in [28] where it was pointed out that an isolated,rigid Euclidean D3-brane instanton wrapping a four-cycle Dinst ofO(1) type (that is,in particular invariant under orientifold action) can generate these missing Yukawacouplings. This requires extra fermionic charged matter zero modes localised atthe intersection of the instanton with the two stacks of D7-branes. Concretely, a

necessary condition is that the chiral intersection numbers are

Ia,inst = 1, Ib,inst = 1 . (45)The resulting six chiral zero modes ia, b, i = 1, . . . 5, can then be absorbed by the

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disc diagrams

10(2,0) iaja, 10

(2,0) kala, 5

(1,1)H

ma b . (46)

As detailed [28] this results in non-perturbative couplings proportional to

Y Yijklm 10ij 10

kl 5

mH (47)

with i , j , . . . denoting SU(5) group indices and , labelling families. Note that thecoupling eq. (47) is of unit rank in family space so that a single instanton gives riseto masses only for one particular generation of u-quarks. This is a consequence of thefact that the multiplicities of the i-modes are only due to their SU(5) Chan-Patonfactors. As stated above, this is a similar situation to the one for the perturbative

couplings, which to first order are also of rank one. For non-perturbative couplings ofthe form (47) the resolution has to involve several distinct instantons whose combinedeffect may be to give rise to higher rank couplings.

Of course the amplitude is suppressed by the instanton action

exp

1

2gs

Dinst

J J

. (48)

As such the scale of the coupling is independent of the GUT coupling, which iscontrolled by the cycle volume of the GUT brane. The instanton cycle entering theabove suppression, however, is a priori unrelated to the GUT cycle. Still in theperturbative regime gs 1 there is the danger that the coupling tends to be toosmall. In our approach we will eventually take the small Tinst limit of the orientifoldmodel and, besides imposing the tadpole constraints, will require that at least for thethird family this Yukawa coupling is generated by a Euclidean D3-brane instanton.

Of course we have to ensure that when Tinst 0 not the whole manifold degenerates.In principle the large hierarchy in the u-quark masses between the third and the firsttwo families can be engineered by different instantons with suitable suppression.

Very similarly, if the Higgs fields originate from the intersection of the SU(5)branes with a third stack then the bottom Yukawa couplings carry U(1)3 charges

10(2,0,0) 5(1,1,0)

5H(1,0,1)

(49)

and are therefore not gauge invariant any more. In this case, also these couplings canonly be generated non-perturbatively by an appropriate D3-brane instanton, whichintersects the U(1) stacks b and c just once.

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Dim=4 proton decay

In GUT theories there is the danger of generating dimension-four operators violatingbaryon or lepton number

U D D, Q D L, L L E . (50)

Clearly if present they generate unacceptably fast proton decay. In Georgi-GlashowSU(5) all these operators descend from the

10(2,0) 5(1,1)

5(1,1)

, (51)

coupling, which is perturbatively forbidden due to U(1)b violation. However, as justdescribed, even perturbatively absent couplings can be generated non-perturbativelyby D3-brane instantons. In certain domains of the Kaahler moduli space suchinstanton-induced dimension-four operators would be dangerous. For an instantonto generate such a coupling three situations are in principle conceivable in view of the

U(1)a and U(1)b charges: it either carries the six charged matter zero modes ia, j

a,

k

b with i,j,k = 1, 2 or alternatively the four zero modes a, a, k

b with k = 1, 2.

The third possibility is that it carries just the two zero modes k

b with k = 1, 2. Onthe other hand, charged matter zero modes from intersections of the instanton with

the SU(5) stack always appear in multiples of five. We thus conclude that in absenceof any known mechanism to absorb the extra zero modes without pulling down morecharged matter fields, in the two first cases no such dangerous dimension-four oper-ators are generated. However, the third option is not a priori excluded. Of course, if

such an instanton exists the coupling is exponentially suppressed, but we just learntin the context of the 10105H Yukawa coupling that this need not be the case in theTinst 0 limit. Therefore, to be on the safe side we require that such an instantondoes not exist.

Neutrino masses

We have already seen that the Yukawa coupling 1N(0,2) 5

(1,1)5H

(1,1) generatesDirac type masses for the neutrinos. In order to realise for instance the see-sawmechanism one also needs Majorana type masses of the order 1012 1015GeV. Thesecan be either generated by higher dimensional couplings involving some additional

SU(5) singlet fields or by D3-brane instantons [25, 26, 64, 65]. Higher dimensionalcouplings are of course suppressed by the string scale, so that one needs to explainthe high scale of these terms.

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For directly generating a mass term

Smass = MN1(0,2)N 1

(0,2)N (52)

via an instanton, it has to carry the four charged matter zero modes ib, i = 1, . . . , 4.In this case the Majorana mass scale is Ms exp(Sinst) which, that is, depending onthe size of the four-cycle, can still give a suppression by a few orders of magnitude.

3.2 Breaking SU(5) to SU(3) SU(2) U(1)Let us describe how one can break the SU(5) GUT via a line bundle LY whosestructure group is embedded into U(1)Y. This method was used in the contextof local F-theory GUT models in [16]. Here we will discuss its implementationwithin the perturbative orientifold and find important quantisation constraints on

the bundle LY . Clearly these have to be taken into account in a string theoreticallyconsistent framework.

Suppose we have designed the model such that the SU(5) gauge symmetry issupported on a D7-brane wrapping a rigid divisor, which is a del Pezzo surfacedPr Y containing r + 1 homological 2-cycles. Therefore, even though we cannotturn on (discrete) Wilson lines (as 1(D) is vanishing), we have the chance to breakthe SU(5) gauge symmetry to the Standard Model by turning on non-vanishing flux

in U(1)Y . This Abelian flux FY is embedded into the fundamental representation ofSU(5) as FY TY SU(5) with

TY =

2 2

23

3

. (53)Such gauge flux through a non-trivial 2-cycle in Y would lead, via the Green-Schwarzmechanism, to a mass term by mixing with an axion. However for flux supported on atwo-cycle of the dPr trivial in Y, there is no axion to pair with and the U(1)Y remainsmassless after gauge symmetry breaking [30,16]. As discussed in Subsection 2.1, thismeans that for U(1)Y to remain massless we have to choose U(1)Y such that its first

Chern class c1(LY) H2(Da,Q) is trivial on the ambient Calabi-Yau space Y, thatis, the element dY H2(Da,Q) specifying LY = ODa(dY) must lie in the kernel ofthe pushforward H2(Da) H2(Y). From the relations eq. (5) it is immediately

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two line bundles La and LY and identifies their field strengths with the followingcombinations of generators Ta and TY of the diagonal U(1)a and hypercharge U(1)Y ,

La

Ta, (56)

LY 25

Ta +15

TY.

The analogue of condition eq. (54),

c1(La) B + 12

KDa Z, (57)

c1(La) + c1(LY) B + 12

KDa Z,

leads to c1(LY) Z and in general half-integer quantised La bundles. This agreeswith the fact that all cohomology groups involve integer powers of La and LY . It isimportant to realise that the gauge flux U(1)Y , though being trivial in the cohomol-ogy on Y nevertheless does contribute to the D3-tadpole condition. The contribution

of the fluxes La and LY reads

Ngauge = 52

Da

c21(La) 15Da

c21(LY), (58)

where we have taken into account tr(T2Y) = 30. Redefining as above La = La L25Y

and LY = L15Y yields

Ngauge =

5

2 Da c21(

La)

Da c21(

LY)

2Da c1(La) c1(LY) . (59)LY being trivial on the Calabi-Yau, the mixed term c1(La) c1(LY) may be non-

vanishing only if also La has components trivial on Y.Let us discuss the effect ofLY in more detail:

Massless U(1)Y

Since we have now embedded the line bundle LY into a combination of T0 and TY,one might be worried that due to the Green-Schwarz mechanism it is not directlyU(1)Y which remains massless. To find the massive Abelian gauge symmetry, we

have to evaluate the relevant axion couplingR1,3Y

C4 Tr(F2GUT), (60)

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where FGUT is the total U(5) field strength supported on the GUT brane stack.Splitting this into the four-dimensional parts F4D and the internal background valuesgiven by the first Chern classes, we can write

FGUT = T0F4D0 +c1(La)B+ 12c1(KDa)+ 25 c1(LY)+TY F4DY + 15c1(LY) . (61)

Inserting this into (60) and extracting the relevant term with two legs of Tr(F2GUT)in the four-dimensional Minkowski space and two legs on the GUT D7-brane, oneimmediately realises that it is still the diagonal F4D0 which mixes with the axions.

Exotics

As already described the decomposition of the adjoint ofSU(5) yields massless statescounted by the cohomology groups H(Da,

LY). For phenomenological reasons we

require that these cohomology groups vanish. This gives already a very strong con-straint on the possible line bundles.

MSSM matter fields

Using the bundles La and LY , we now express the relevant cohomology groups count-ing the number quarks and lepton fields. As mentioned these modes localise eitheron surfaces or on curves. To treat both cases simultaneously we express the numberof modes in terms of sheaf extension groups. It is understood that for the actualcomputation one uses the formulae collected in Subsection 2.3.

The anti-fundamental matter representation of SU(5) splits as

(3, 1)2Y : Ext(La, L1b ), (62)

(1, 2)3Y : Ext(La LY, L1b )

with Lb = Lb. Similarly, for the anti-symmetric representation, we have to computethe three cohomology classes

(3, 2)1Y : Ext(L1a L1Y , La),

(3, 1)4Y : Ext(L1a , La), (63)

(1, 1)6Y : Ext(L1

a L2

Y , La).Since LY is trivial in Y it is guaranteed that the chiral index of these representationsdoes not change. However, in general the vector-like matter will change and will

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be different for MSSM states descending from the same GUT multiplet. This isavoidable if all matter is localised on curves (and not on surfaces) and that therestriction ofLY to this matter curve vanishes.

Higgs field and 3-2 splitting

The fundamental representation for the Higgs field of SU(5) splits as

(3, 1)2Y : Ext(L1a , L1b ), (64)

(1, 2)3Y : Ext(L1a L1Y , L1b ).

Note that all these states are vector-like. We need that the SU(2) doublets remain

massless and that the SU(3) triplets get a mass of the GUT scale. This translates intorequiring that Ext(L1a , L1b ) = (0, 0, 0, 0) and Ext(L1a L1Y , L1b ) = (0, 1, 1, 0).For the Higgs fields localised on the intersection curve Cab of the U(5) divisor Daand the U(1) divisor Db, two possibilities can occur.

The first option is that the intersection locus is a single elliptic curve, that is,Cab = T2. In this case the restriction of the line bundles to Cab have to be of degreezero so that indeed no chiral matter is localised on Cab. Recall that a degree zero linebundle on the elliptic curve Cab can be written as O(p q), where p, q denote pointsdifferent from the origin 0 of the elliptic curve. The trivial bundle O correspondsto p q = 0 and has cohomology H(Cab, O) = (1, 1). Ifp q = 0 the line bundlehas a non-trivial Wilson line and the cohomology groups vanish. It is therefore clearthat for an appropriate choice of the line bundles appearing in eq. (64) it can be

arranged that only the doublet remains massless while the triplets acquire stringscale masses. According to what we just said this happens provided the restrictionof the line bundles appearing in eq. (64) to the genus one matter curve Cab take theform

L1a Lb|Cab = O(p q), p q = 0,L1a L1Y Lb|Cab = O. (65)

To see how to arrange for this, suppose one has found a model where the line bundlesLa and Lb can both be written as the pullback of line bundles from the Calabi-Yau,

La = a La, Lb = b Lb. (66)Since L1a Lb|Cab is of degree zero in this case L1a Lb|Cab = O with trivial Wilsonlines. The relations eq. (65) can now simply be met by twisting La by a line bundle

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Ra on Da which is trivial on the ambient manifold and satisfies

Ra = L1Y . (67)

As with everything desirable in life this surgery is not for free, as the new contributionof the GUT brane to the D3-brane tadpole is increased from eq. (59) to15

Nagauge = 5

2

Da

c21(La) 3

2

Da

c21(LY). (68)

We will however also encounter cases16 in which eq. (66) is not the situation tobegin with. In particular it may be inconsistent to define the final GUT bundleas aLa L1Y because this bundle might exhibit ghosts in its spectrum. For theline bundle appearing in (64), in this case we have, supposing for definiteness that

Lb = bLb,

L1a L1Y Lb|Cab = O(p1 q1). (69)We then need to twist Lb by a line bundle Rb trivial on the ambient space, and everysuch bundle satisfies Rb|Cab = O(p2 q2)|Cab. To ensure p1 + p2 (q1 + q2) = 0,as desired, might require adjusting some of the complex structure moduli of themanifold.

On the other hand, it was argued in [16] that in case Hu and Hd are localisedon a single curve, dimension five proton decay operators Q Q Q L can be generatedby exchanging Kaluza-Klein modes of the Higgs-triplet. To avoid such operators,it was suggested that the intersection locus Da

Db consists of two components

C1 C2, such that the 5H originates from H(C1, L1a Lb KC1) = (1, 0) and5H from H

(C2, L1a Lb

KC1) = (0, 1). This is the second option we have forthe localisation of the Higgs field.

The top Yukawa couplings

We have seen that in the SU(5) GUT model the top quark Yukawa coupling 10105Hcan be generated by a single rigid O(1) instanton with the right charged matter zero

mode structure. We need to check whether this is compatible with the breaking ofthe SU(5) GUT group by the U(1)Y flux.

15Here we are using that La = a La prior to twisting by Ra so that the cross-term in eq. (59)vanishes.

16This discussion is only relevant for the models proposed in Section 5 and Subsection 6.1.

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Recall that the five zero modes ia transform in the 5 representation of the SU(5)and the single zero mode b in the singlet representation ofSU(5) with U(1)b chargeqb = +1. The U(1)Y flux splits the 5 representation according to (55), that is,

the (3, 1)2Y zero modes are counted by Ext(La, O) and the (1, 2)3Y modes byExt(La LY, O). As long as the SU(5) stack of branes and the instanton braneintersect over a 2-cycle non-trivial in the Calabi-Yau manifold, the restriction of LYto the intersection locus vanishes and we get precisely three instanton zero modesia transforming in the (3, 1)2Y representation and two zero modes

ja transforming

in the (1, 2)3Y representation. Since LY is supported on the SU(5) stack, thesingle zero mode b also still exists. Then the Standard Model top-Yukawa couplings(3, 2) (3, 1) (1, 2)H are generated by the instanton via the following absorption of thesix charged instanton zero modes

(3, 2)1Y (3, 1)a2Y (1, 2)a3Y(3, 1)4Y (3, 1)a2Y (3, 1)a2Y (70)(1, 2)3Y

(1, 2)a3Y (1, 1)b0Y .The -term

The supersymmetric term clearly vanishes at tree-level. For having it directlygenerated non-perturbatively, the rigid O(1) instanton must carry the four chargedmatter zero modes a, a and b, b. However, due to the SU(2) Chan-Paton factorthe always come in multiples of two, so that these simple non-perturbative -

terms are absent. However, they can be generated by higher dimensional operatorsinvolving SU(5) gauge singlets, which have to receive some non-vanishing vacuumexpectation value.

3.3 Summary of GUT Model Building Constraints

In this section we have collected a number of perturbative and non-perturbativestringy mechanisms to first realise Georgi-Glashow SU(5) GUTs and second to solvesome of their inherent problems. The perfect string model, besides being globallyconsistent would of course satisfy all these constraints. Eventually, one also has to

address the issue of moduli stabilisation by fluxes and instantons, some aspects ofwhich we discuss in Section 8. A more thorough and complete analysis is beyond themain scope of this paper but truly on the agenda.

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the surfaces dPn in Subsection 4.1 and classify all involutions on these surfaces. InSubsection 4.2 we argue that the elliptically fibred Calabi-Yau threefolds over dPncan also be obtained by del Pezzo transitions starting from the degree 18 hypersurface

in P1,1,1,6,9 after flop transitions. Simple examples are obtained if the base is oneof the toric del Pezzo surfaces dP2 or dP3. In Sections 4.3 and 4.4 we constructthe corresponding Calabi-Yau threefolds and study their different topological phasesusing toric geometry. In a second step we introduce viable orientifold involutions on these compact Calabi-Yau manifolds and derive the induced tadpoles from O7and O3 planes.

In the second part of this section we will have a closer look at the topologicalphases ofMn with n dP8 surfaces. We show in Subsection 4.5 that these are examplesof so-called swiss-cheese Calabi-Yau manifolds which can support large volume vacuawith one large and n + 1 small four-cycles [7, 8, 66]. In Subsection 4.6 we discuss the

D-term conditions arising from wrapping a D7-brane on the small del Pezzo surfaceswith orientifold invariant homology class.

4.1 Del Pezzo Surfaces and Their Involutions

The compact orientifold geometries for our GUT models will be obtained from ellipticfibrations over del Pezzo surfaces dPn. In order to study these threefolds it will benecessary to review some basic facts about del Pezzo surfaces first. We will alsodiscuss involutions on these dPn. We will determine their fix-point locus and actionon the exceptional curves of the del Pezzo surface. Since these involutions on the

base will descend to involutions on the entire Calabi-Yau manifold, this will enableus to identify viable brane configurations later on.

On the geometry of del Pezzo surfaces

By definition, del Pezzo surfaces are the Fano surfaces, that is, the algebraic surfaceswith ample canonical bundle. These are either the surfaces Bn = dPn, which areobtained by blowing up P2 on 0 n 8 points17, or P1 P1. Their Hodge numbers

17The points must be general in the sense that no two points are infinitesimally close, no threeare on one line, no six on a conic, no eight on a cubic with a node at one of them. In other words,one is not allowed to blow up points sitting on a (

1)-curve. If one were to blow up a point on a

(1)-curve, then the proper transform would be a (2)-curve. So yet another characterisation ofthe allowed points is that there be no curves of self-intersection 2 or less. Moreover, note thatdifferent sets of points can correspond to the same (complex structure on the) del Pezzo surface.

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are h0,0 = h2,2 = 1, h1,1Bn = n + 1 and h1,1P1 P1 = 2, while all other hp,q

vanish. A basis of homologically nontrivial two-cycles in Bn consists of the class ofa line l in P2, and the n exceptional curves ei, one for each blown-up point. Their

intersection numbers are l2

= 1, ei ej = ij, ei l = 0. Written in this basis, thefirst Chern class isc1

TBn

= K = 3l ni=1

ei . (71)

The square of the canonical class

K2 =

Bn

c21 = 9 n (72)

is also called the degree18 of the del Pezzo surface. The second (top) Chern class isthe Euler density, hence

Bn = Bn

c2 = 3 + n . (73)

Let C be a curve in the del Pezzo surface. Then its degree deg(C) and its arithmetic

genus g read

deg(C) = K C , g = 12 (C C+ K C) + 1 . (74)

Of particular interest are the rigid genus-0 instantons, that is rational curves of self-intersection (1). For convenience of the reader we reproduce the classification ofsuch (1)-curves, see [67], in Table 4.

The del Pezzo surfaces P2 = B0, P1 P1, B1, B2, and B3 (that is, those ofdegree K2 6) are toric varieties. The remaining surfaces B4, . . . , B8 are not toricvarieties, but can of course be embedded into toric varieties. In particular, the delPezzo surfaces B5, . . . , B8 are hypersurfaces or complete intersections in weightedprojective spaces. For this, let us denote by P(d1, . . . , dr|w0, . . . , wm) the completeintersection of r equations of homogeneous degree d1, . . . , dr in weighted projectivespace with weights w0, . . . , wm. These del Pezzo surfaces are listed in Table 5. Oneinfers that the dimension of the complex deformation spaces for del Pezzo surfacesBn with n 5 is dim H1(TBn) = 2n 8.

18To understand this notation, note that a degree d = K2 del Pezzo surface with d

3 can be

realised as a degree-d hypersurface in Pd.

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class H2(Bn,Z) B1 B2 B3 B4 B5 B6 B7 B8(0; 1) 1 2 3 4 5 6 7 8

(1; 12) 1 3 6 10 15 21 28(2; 15) 1 6 21 56

(3; 2, 16) 7 56(4; 23, 15) 56(5; 26, 12) 28(6; 3, 27) 8

Total no. 1 3 6 10 16 27 56 240

Table 4: Number of (1)-curves on the Bn del Pezzo surfaces. Thecoefficients (a; b1, . . . bn) are with respect to the standard basis(l; e1, . . . , en). For example, (1, 12) denotes all

n2

homology

classes of the form l ei ej, 1 i < j n. Note that thereare no (1)-curves on B0 = P2 andP1 P1, which are omitted.

del Pezzo K2 hypersurface coordinates

B5 4 P(2, 2|1, 1, 1, 1, 1) (x1, x2, x3, x4, x5)B6 3 P(3|1, 1, 1, 1) (x1, x2, x3, x4)B7 2 P(4|2, 1, 1, 1) (y, x1, x2, x3)B8 1 P(6|3, 2, 1, 1) (y , z , x1, x2)

Table 5: The del Pezzo surfaces of degree K

2

4.

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Classification of involutions on del Pezzo surfaces

In order to systematically study GUT models on elliptically fibred Calabi-Yau mani-folds with del Pezzo base, one needs to classify all different, non-trivial, holomorphicinvolutions on del Pezzo surfaces. In the following we intend to discuss the finalclassification in Table 6 and give a first impression of the necessary steps needed forthis derivation. Most of the technical details and geometric constructions are shiftedinto Section A. The classification in Table 6 completes the list obtained in ref. [41].

For a systematic classification of involutions we will look at the pattern of rigidP1 instantons, that is, the (1)-curves. Clearly, every involution induces a Z2 auto-morphism of the (1)-curves. Conversely, up to degree 6, the automorphism of the(1)-curves determines the involution. In the remaining degrees 7 there eitherare no (1)-curves, or they lie over a line or point of the blown-up P2. Hence, inthe latter case they do not fill out the space to uniquely determine the involution.Technically, the (1)-curves generate all of H2(S,Z) if and only if the degree is 6or less. In a next step, one has to find all conjugacy classes of involutions actingon the (1) curves and check that these descend to actual geometric involutions onthe corresponding del Pezzo surface. The details of this analysis can be found inSection A. Here we will discuss the final classification summarised in Table 6.

In Table 6 the complete list of del Pezzo surfaces Bn with involutions i is shown.For each pair (Bn, i) it also includes detailed information about the fix-point set.We use the following notation:

() is the homology class of the genus-g curve with g

1 in the fixed point

set of. As discussed in Section A, there is at most one such curve.

R() are the homology classes of the rational genus 0 curves in the fixed pointset.

B() is the number of isolated fixed points that do not lie on (1)-curves. P() is the number of isolated fixed points that do lie on (1)-curves, and

hence may not be blown up further.

(b+2 , b2 ) are the dimensions of the eigenspaces of i in H2(Bn).

In the last column we also displayed the explicit action of the involution on the basis(l, e1, . . . , en) ofH2(Bn) and basis (l1, l2) ofH2(P1 P1). The k-dimensional identitymatrix is simply denoted by 1k, while H exchanges two elements ei ej or l1 l2.

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In addition there are also five involutions (I(2)B3 , I

(3)B3 , I

(5)B5 , I

(9)B7 , I

(9)B8 ) which should be

viewed as the additional building blocks for all non-trivial involutions on del Pezzosurfaces. We will introduce the explicit form of these involutions in turn.

To define the special involutions we will specify their action on the basis elements(l, e1, . . . , en). On the third del Pezzo surface we define the two involutions

I(2)B3 =

2 1 1 11 1 0 11 0 1 11 1 1 0

, I

(3)B3 =

2 1 1 11 0 1 11 1 0 11 1 1 0

. (75)

The remaining three involutions we need to introduce are well-known classical in-volutions. They are minimal since they satisfy (E) = E and (E) E = foreach (1)-curve E. This implies that such an involution cannot be obtained byblowing up a higher degree del Pezzo and extending an involution defined on the delPezzo surface before blow-up. On B5 there is a minimal involution known as the deJonquieres involution which acts as

I(5)B5 =

3 2 1 1 1 12 1 1 1 1 11 1 1 0 0 01 1 0 1 0 01 1 0 0 1 01 1 0 0 0 1

. (76)There is one minimal involution for the del Pezzo surfaces of degree 1 and 2, respec-tively. The del Pezzo surface B7 admits the Geiser involution

I(9)B7 : l l 3K , ei K ei . (77)

while on B8 one has the Bertini involution acting asI

(9)B8 : l l 6K , ei 2K ei . (78)

Note that one can explicitly check that each involution on each del Pezzo surfacepreserves its canonical class K defined in eq. (71).

With these definitions at hand, the involutions in Table 6 can be used for ex-plicit computations. This will be particularly useful for the elliptically fibred three-folds over a del Pezzo base, since all involutions can be lifted to the correspondingCalabi-Yau threefold. We are then in the position to construct explicit Calabi-Yauorientifolds and compute the tadpoles induced by the O3- and O7-planes.

4.2 The Geometry of Del Pezzo Transitions ofP1,1,1,6,9[18]

We are now in the position to construct compact Calabi-Yau threefolds Mn associatedto a del Pezzo base. The first construction is via an elliptic fibration over a del Pezzobase Bn, while the second construction is by employing del Pezzo transitions.

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Elliptically fibred Calabi-Yau threefolds with del Pezzo base

Let us construct Calabi-Yau threefolds Mn as elliptic fibrations over the del Pezzobase dPn. We consider elliptic fibrations which are generically smooth with the worstdegeneration of the fibre of Kodaira type I1 [68,69]. In the following we will restrictfurther to elliptic fibrations with generic elliptic fibres of type E8 such that thegeneric elliptic fibres can be represented by a degree 6 hypersurface in P1,2,3 denotedby P1,2,3[6]. As shown, for example, in [70], one then finds that the Euler number ofthe elliptic fibration Mn is given by

(Mn) = C(8)Bn

c21 = 60(n 9), (79)

where C(8) = 30 is the dual Coxeter number of E8 and we have used (72) for the delPezzo base

Bn. One can also count the number of Kahler classes for these geometries.

One finds that there are n + 1 classes corresponding to the non-trivial two-cycles ofthe del Pezzo base as well as the fibre class of the elliptic fibration. This implies thatMn has Hodge numbers

h1,1(Mn) = n + 2 , h2,1(Mn) = 272 29n, (80)

where we have used that = 2(h1,1 h2,1).The specification ofMn as an elliptic fibration over the base dPn will turn out to

be particularly useful in the study of orientifold involutions and brane configurationson Mn. Let us introduce the map

: Mn Bn , (81)which is the projection from the threefold Mn to the base. Note that every (1)curve class E in Bn can be pulled back to a divisor in Mn using : E (E) H4(Mn,Z). In fact, each such divisor is a dP9 surface. This surface is defined as blowup ofP2 at nine points which arise at the intersections of two cubic curves. Thus, dP 9is an elliptic fibration over P1 which has 12 singular fibres19. The dP9 is not strictly

a del Pezzo surface, but the equations (72) and (73) remain to be valid. Recall thatthe (1) curve in the base have been listed in Table 4. It it thus straightforward todetermine the intersections of these curves. In case two curves E1, E2 intersect at

a point the corresponding two dP9 divisors (E1) and (E2) in Mn will intersect19Roughly speaking, the dP9 is half a K3 surface which is an elliptic fibration over P

1 with 24singular fibres.

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over a Riemann surface of genus 1. Clearly each (E) intersects the base Bn in aP1. Already this simple analysis allows us to infer the necessary information on thetriple intersections of the threefold Mn in the elliptically fibred phase.

We want to apply a similar logic also for the extension of an involution on thedel Pezzo base to an involution on the threefold Mn. In fact, by appropriatelydefining the action on the (1) curves ofBn lifts to an action of their pull-backs.The fixed divisors wrapped by the O7-planes can then be determined using Table 6.Determining the number of O3-planes in the full set-up also depends on the preciseform of the involution on Mn. In particular, it is not generally the case that each

isolated fix-point in the base lifts to a single fix-point in Mn. Let us consider the casewhere the torus fibre over the fix-point is s

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Ralph Blumenhagen et al- GUTs in Type IIB Orientifold Compactifications

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